Origin of the Fine Structure of the Raman $D$ Band in Single-Wall Carbon Nanotubes

Experiments show that the $D$ bands of bundles of single wall carbon nanotubes have a fine structure, apparently consisting of more than one subband. Using the double resonance theory, we calculate for the first time the $D$ band for a sample of a given diameter distribution for seven different laser excitation energies in a wide range. In addition, a detailed theoretical explanation for the fine structure of the $D$ band is provided. The calculated results agree well with experiments and show that the main factors in determining the fine structure are an enhanced trigonal warping of the phonon dispersion, the presence of a diameter distribution in the sample, and—most importantly—the resonance from the Van Hove singularities.

Figure 1

Measured (dotted curve) $D$ band for $E_{\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{r}}=1.761\mathrm{e}\mathrm{V}$; calculated results using the model phonon dispersion of Eq. (1) with parameters ${\omega }_1=120{\mathrm{c}\mathrm{m}}^{-1}$ and $\delta =0.06$ (thin solid curve); calculated results using the model phonon dispersion of Eq. (1) with parameters ${\omega }_1=170{\mathrm{c}\mathrm{m}}^{-1}$ and $\delta =0.3$ (thick solid curve); and calculated results using the DFT phonon dispersion of Kresse and Dubay (dashed curve). The frequencies of the calculated spectra have been scaled up relative to the value of $1220{\mathrm{c}\mathrm{m}}^{-1}$ for the low (thin solid) and the high (thick solid) anisotropy model dispersion calculations. DFT calculation (dashed) was not scaled. The inset shows the equi-phonon-frequency (EPF) contours around the $K$ point for the high anisotropy model phonon dispersion compared with the EPF contours of the DFT phonon dispersion of Kresse and Dubay (dashed curves) for $1300{\mathrm{c}\mathrm{m}}^{-1}$ (inner curves) and $1350{\mathrm{c}\mathrm{m}}^{-1}$ (outer curves). The numbers on the axes are the dimensionless wave vectors in units of $1/(\sqrt{3}{a}_0)$, where $a_0=0.144\mathrm{n}\mathrm{m}$ is the C-C bond length.

Figure 2

Measured results (dotted curve), calculated results using the model phonon dispersion of Eq. (1) with parameters ${\omega }_1=170{\mathrm{c}\mathrm{m}}^{-1}$ and $\delta =0.30$ (solid curve), and calculated results using the DFT phonon dispersion of Kresse and Dubay (dashed curve) for the $D$ band for six different values of $E_{\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{r}}$. The model calculated spectra were scaled by the following values in increasing order of $E_{\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{r}}$: $0\%$, $1\%$, $1\%$, $6\%$, $12\%$, and $13.5\%$. For the DFT calculated spectra, these values were $0\%$, $0\%$, $0\%$, $0\%$, $16\%$, and $18\%$.